Problem: If $a$ is the smallest positive two-digit multiple of $3$, and $b$ is the smallest positive three-digit multiple of $4$, then find $a+b$.
Explanation: Dividing 10 by 3 gives a quotient of 3 and a remainder of 1. Therefore, $3 \cdot 3$ is the largest one-digit multiple of 3, and $3 \cdot 4$ is the least positive two-digit multiple of 3. This calculation shows that $a=12$.

Dividing 100 by 4 gives a quotient of 25 and no remainder. Therefore, $4 \cdot 25$ is the least positive three-digit multiple of 4, and $b=100$.

Combining these results gives $a+b = 12+100 = \boxed{112}$.